The Mi ro-Meso
onne tion
also known as
Non-Equilibrium Statisti al Me hani s
also known as
The Theory of Coarse-Graining
Pep Español
Le ture notes at Courant, November 2013
Contents
1 The Mi ros opi Dynami s
9
1.1
Classi al Me hani s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Hamilton's equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
Liouville's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
level
9
1.4
Equilibrium at the mi ros opi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.5
The single, all en ompassing problem of Non-Equilibrium Statisti al Me hani s . . . . . .
17
2 The Mesos opi Dynami s
2.1
Levels of des ription
2.2
Sto hasti
2.3
2.4
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
pro esses in Phase Spa e
19
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Green's view of
oarse-graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Zwanzig view of
oarse-graining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P (a, t)
29
2.4.1
Exa t equation for
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4.2
The Markovian approximation and the Fokker-Plan k Equation . . . . . . . . . . .
30
3 Example: Diusing intera ting olloidal parti les
35
8
CONTENTS
1
The Mi ros opi Dynami s
The s ope of these notes has some restri tions.
In parti ular, we will
We will assume also that at the most mi ros opi
level the system
me hani s or, to be more spe i , by Hamilton's equations.
be made.
Another
onsider only isolated systems.
an be well des ribed by
lassi al
No referen e to quantum me hani s will
ru ial assumption is that the Hamiltonian dynami s of the system has a well-
dened equilibrium state that is rea hed by the system as the time pro eeds. The assumption of isolated
system implies that we will look at the relaxational dynami s of the system towards its equilibrium state.
This might seem a strong restri tion from an experimental point of view. Experiments often deal with
situations in whi h a system is subje t to the a tion of external inuen es, usually through the boundary
of the system. Nevertheless, the theory for isolated systems already provides the basi
to whi h boundary
1.1
onditions
model equations
an be applied in a latter stage.
Classi al Me hani s
We will deal with ma ros opi
systems that
an be appropriately des ribed with Classi al Me hani s.
Classi al Me hani s is a theory of point parti les with denite positions
through for es.
ri
and velo ities
vi
that intera t
We will often idealize and refer to atoms and even mole ules as point parti les, even
though they may be
omposite obje ts. The fun tional forms of the for es between the point parti les is
known only in an approximate way, and model for e elds are usually required and adopted. In prin iple,
these for es elds should be derived from the quantum me hani al origin of the atoms or mole ules that
are represented at a
lassi al level with point parti les. The formulation of a
subje t in itself of great
1.2
urate for e elds is a vast
urrent interest.
Hamilton's equations
There are many dierent formulations of the laws of Classi al Me hani s whi h are all equivalent to the
original formulation set forth by Newton. Newton's Laws give rise to equations of motion for the positions
of
lassi al obje ts that are diferential equations of se ond order. Their solution requires the knowlegde
of the initial
onditions given by the positions and velo ities in order to predi t the future evolution of the
system. As stressed originaly by Gibbs, the Hamiltonian des ription of Classi al Me hani s is parti ularly
suited to the formulation of Statisti al Me hani s be ause one of the distinguishing features of Hamilton's
equations is that they are rst order dierential equations. In Hamilton's formulation, the mi ros opi
state
z = {qi , pi }
of a system of
N
point parti les is given by the
olle tion of all
qi
and momenta
pi
of
The Mi ros opi
10
the parti les. The mi ros opi
Dynami s
state of the system evolves a
ording to Hamilton's equations
∂H
(z),
∂pi
∂H
(z),
ṗi = −
∂qi
q̇i =
where
H(z)
(1.1)
(1.2)
is the Hamiltonian fun tion assumed to be expli itly independent of time.
Typi ally the
Hamiltonian has the form
H(z) =
where the rst sum is the kineti
is
U (r1 , · · · , rN )
and
V (r)
N
N
X
X
p2i
V (ri )
+ U (r1 , · · · , rN ) +
2m
i=1
i=1
energy of the sytem, the potential of intera tion between parti les
is a time-independent external potential.
Hamiltonian is a bounded fun tion from below.
potential fun tion without
The
(1.3)
Be ause we
an
We will always assume that the
hoose an arbitrary
hanging the dynami s, we will assume that
H(z) ≥ 0
onstant for the
for all mi rostates
z.
ondition of bounded Hamiltonian is a requisite for the existen e of a proper equilibrium state. One
important
ase for whi h this
potential. In this
ondition is not satised is when the system intera ts with a gravitational
ase there are mi rostates for whi h the energy is arbitrarily negative (as when two
massive point parti les keep approa hing and de reasing without bound its potential energy). A
olle tion
of self-gravitating point obje ts does not have a well-dened equilibrium state.
Hamilton's equations are rst order dierential equations that require the knowledge of an initial
ondition
z0 = {qi (0), pi (0)}.
The ulterior evolution of the mi rostate
dimensional spa e of all mi ros opi
states
z
known as the phase spa e
zt is a traje tory in the 6N
Γ of the system. In fa t, the
traje tory is restri ted to live in a submanifold of the full phase spa e, be ause of the existen e of
dynami invariants.
A dynami al invariant
I(z)
is any dynami al fun tion that does not
hange in
time, this is,
d
I(z(t)) = 0.
dt
The traje tory
z(t)
(1.4)
is, therefore, restri ted to be in the submanifold
I(z) = I0
where
I0 = I(z0 )
is the
value of the dynami al invariants at the initial time. The dynami al invariants emerge as the result of
symmetries of the Hamiltonian. A
ording to a fundamental theorem due to Emmy Noether, to every
symmetry of the Hamiltonian there is
that the Hamiltonian itself is a
momentum is
onserved property.
Invarian e under time translation ensures
onserved quantity. Invarian e under spa e traslations ensure that total
onserved, while rotation invarian e ensure that angular momentum is
independent Hamiltonian like (1.3)
onserved. A time
onserves energy. If there is no external potential and the intera -
tion potential depends only on the relative distan es between parti les, then total linear and angular
momentum will also be
onserved. For typi al Hamiltonian systems des ribing mole ular systems, the
traje tories in phase spa e display the phenomenon of
we start with initial
haos. This term refers to the property that if
onditions that are very similar, the traje tories starting at these initial
onditions
separate from ea h other exponentially in time. Therefore, while Hamilton's equations are deterministi ,
in pra ti e its predi tive power is rather limited be ause any small un ertainty in initial
explodes and renders the predi tion very ina
onditions readily
urate. Although this seems to be an unfortunate feature, it
is in fa t what makes the statisti al methods appli able and what, in the last instan e, makes Statisti al
Me hani s a predi tive theory.
Hamilton's equations may be writen in
ompa t form
żt = J0
∂H
(zt )
∂z
(1.5)
1.2 Hamilton's equations
J0
is the so
alled symple ti
11
matrix having a blo k diagonal matrix form with the blo ks given by
0 1
−1 0
.
(1.6)
We may write also Hamilton's equations in the form
ż = iLz
where
iL
(1.7)
is the Liouville operator that has the expli it form
N X
∂H ∂
∂H ∂
∂H ∂
iL ≡
−
J0 .
=−
∂pi ∂qi
∂qi ∂pi
∂z ∂z
i
(1.8)
F (z)
an be expressed in terms of the Poisson
iLF (z) = −{H, F }
(1.9)
The Liouville operator a ting on an arbitrary fun tion
bra ket,
where the Poisson bra ket of two arbitrary fun tions
{F, G} ≡
Given an initial mi rostate
z,
F (z), G(z)
is
∂F ∂G
∂F ∂G
−
∂qi ∂pi
∂pi ∂qi
the solution of Hamilton's equations
(1.10)
an be denoted by
zt = T t z
(1.11)
Tt is a one-parameter operator a ting on the initial ondition z . This operator satises T0 = 1 and
Tt Tt′ = Tt+t′ . A formal expression for this operator an be obtained in terms of the Liouville operator.
where
Indeed, the formal solution of (1.7) is
zt = exp{iLt}z
(1.12)
where the exponential operator is dened formally through the Taylor series
exp{iLt} ≡ 1 + iLt +
1
1
(iLt)2 + (iLt)3 + · · · .
2!
3!
Note that by substitution of (1.13) into (1.12) we re over the usual Taylor series of
z(0) = z .
Any fun tion in phase spa e
dependent mi rostate, this is
X(Tt z).
X(z)
(1.13)
z(t) = zt
around
a quires a time dependen e on e evaluated on the time
Phase fun tions evolve be ause the mi rostates evolve in time.
The time derivative of this fun tion is
∂H
∂X
d
∂X
d
X(Tt z) =
(Tt z) Tt z =
(Tt z)J0
(Tt z) = iLX(Ttz)
dt
∂z
dt
∂z
∂z
(1.14)
This dierential equation has as formal solution
X(Tt z) = exp{iLt}X(z)
(1.15)
The Mi ros opi
12
Dynami s
Time reversibility
Hamilton's equations are time reversible equations. In order to grasp the meaning of this statement, it
is
onvenient to introdu e the time reversal operator dened as the operator
z = {ri , pi } and produ es the mi rostate ǫz = {ri , −pi }, i.e.
2
Obviously ǫ is the identity operator. With this operator, it is
ǫ
that takes a mi rostate
it reverses the sign of the momentum.
straighforward to observe the following
property of the Lioville operator

.
.
.

 rj
X ∂H(z) ∂
∂H(z) ∂ 

iLǫz =
−

∂p
∂r
∂r
∂p
i
i
i
i
 −pj
i

.
.
.


.
.
.
pi
mi


.
.
.
pi
mi

 



 



 



=
 = −ǫ 
 = −ǫiLz
 



  −Fi 
 Fi 
 



.
.
.
(1.16)
.
.
.
This is, the time reversal operator and the Liouville operator anti ommute when applied to mi rostates
iLǫz = −ǫiLz
This is a ree tion of the fa t that the symple ti
matrix
J0
(1.17)
and the time reversal matrix
ǫ anti
ommute,
this is

..
.
.
.
.
.
.
.

 ···
0 1
J0 ǫ = 
 · · · −1 0

.
.
.
···
···
.
.
.
..
.

..
.
.
.
.

 ··· 1

 ··· 0

This property has the following
.
.
.

.
.
.

..
.
.
.
.
.
.
.
 

0 ··· 
0 −1 · · ·
 =  ···
 · · · −1
−1 · · · 
0 ···
 
.
.
.
..
.
.
.
.
.
.
.
..
.



 = −ǫJ0


onsequen e on the evolution operator applied to an initial
(1.18)
ondition
z,
exp{iLt}z ,
1
1
ǫ exp{iLt} = ǫ 1 + iLt + iLiLt2 + · · · z = 1ǫ − iLǫt − iLǫiLt2 + · · · z
2!
2!
1
= 1ǫ − iLǫt + iLiLǫt2 + · · · z
2!
= exp{−iLt}ǫz
Changing the notation to
Tt = exp{iLt}
(1.19)
we have
ǫTt z = T−t ǫz
(1.20)
Tt ǫTt z = ǫz
(1.21)
or, equivalently,
In words, this equation expresses the fa t that if we take a mi rostate
Hamilton's equations to get
Tt z ,
then we evolve the resulting mi rostate for a time t, to get
with the velo ities reversed
ǫz .
z
and let it evolve a
ording with
and then we reverse the velo ities of the evolved mi rostate
Tt ǫtTt z ,
ǫTt z ,
and
we end up with the initial mi rostate
1.3 Liouville's theorem
1.3
13
Liouville's theorem
Hamilton's equation are rst order diferential equations that provide the deterministi
system provided that the initial
ondition
z
evolution of the
is given. However, it is in general impossible to know the
pre ise value of the initial values of the positions and momenta of all the parti les in our system. Usually,
a system is prepared under identi al ma ros opi
onditions that do not allow to x the value of the
positions and momenta of every single parti le in the system. A
olle tion of identi al systems prepared
in identi al manner at the initial time will have, in general, dierent initial mi ros opi
reason the best we
probabilisti
initial
terms by introdu ing a probability density
ρ0 (z)
that the system has the mi rostate
z
as
ondition. The probability distribution in phase spa e is usually referred to as an ensemble. Even
though the evolution of
zt
in phase spa e a sto hasti
by
states. For this
an do is to express our knowledge about the initial mi rostate of the system in
ρt (z)
is deterministi , the un ertainty about initial
onditions renders the evolution
pro ess. The probability distribution fun tion at a subsequent time is denoted
and it obeys the Liouville equation whi h we will derive below for
ompleteness. This equation
is just an expression of a very important property of the solution of Hamilton's equation whi h is the
onservation of volume in phase spa e.
We may think of (1.11) as a
oordinate transformation from
z
to
zt .
The Ja obian of this
oordinate
transformation is, by denition, the following determinant
Jt (z0 ) ≡ det (Jt (z0 ))
(1.22)
where the Ja obian matrix is
Jt (z0 ) ≡
The time derivative of the Ja obian
an be
∂zt (z0 )
∂z0
(1.23)
omputed by using the identity
Jt (z0 ) = det (Jt (z0 )) = exp{Tr[ln Jt (z0 )]}
(1.24)
where the logarithm of the Ja obian matrix is dened in terms of the Taylor series of the logarithm.
One way to prove the above identity is by diagonalizing the Ja obian matrix with an orthogonal matrix.
Be ause the determinant and tra e operations are invariant under su h a transformation, we nd that
exp{Tr[ln Jt (z0 )]} = exp
where
λk
are the eigenvalues of
J.
(
X
k
ln λk
)
=
Y
λk
(1.25)
k
The last identity is just the determinant of the Ja obian matrix. If
we now take the time derivative of the Ja obian, we obtain
d
d
Jt (z0 ) = exp{Tr[ln Jt (z0 )]} Tr[ln Jt (z0 )]}
dt
dt d
−1
= exp{Tr[ln Jt (z0 )]}Tr
Jt (z0 ) Jt (z0 )
dt
d
−1
Jt (z0 ) Jt (z0 )
= Jt (z0 )Tr
dt
(1.26)
The Mi ros opi
14
Dynami s
Consider the time derivative of the Ja obian matrix
d
∂ d
d ∂zt (z0 )
=
Jt (z0 ) =
zt (z0 )
dt
dt ∂z0
∂z0 dt
∂H
∂
J0
(zt (z0 ))
=
∂z0 ∂z
∂2H
∂zt (z0 )
= J0
(zt (z0 ))
∂z∂z
∂z0
∂2H
= J0
(zt (z0 ))Jt (z0 )
∂z∂z
(1.27)
Therefore, by inserting this result into (1.28) we obtain
d
∂2H
Jt (z0 ) = Jt (z0 )Tr J0
(zt (z0 )) = 0
dt
∂z∂z
Be ause the Ja obian is
This result is a spe ial
t= 0
onstant and at
(1.28)
it takes the value 1, then it will always be equal to 1.
?
ase of the theorem of the integral invariants of Poin aré [ ℄). The fa t that the
Ja obian of the evolution is always equal to one has an important
onsequen e on the evolution of the
probability density in phase spa e, as we now show.
Let
M
be a region of not vanishing measure of
ea h point of
in the region
M
M
a
at
t=0
Tt M
and
the region resulting from the evolution of
is identi al to the probability of being at
Z
By performing the
Γ
ording to Hamilton's equations. It is obvious that the probability that the system is
Z
M
at
t = t.
ρt (z)dz.
For this reason,
(1.29)
(with unit Ja obian) the integral in the left hand side
ρt (z)dz =
Z
ρt (Tt z)dz.
(1.30)
M
Tt M
This is true for any region
Tt M
Tt M
z ′ = T−t z
hange of variables
be omes
ρ0 (z)dz =
M
Z
and, therefore, the integrand of the right hand side of (1.29) and the left
hand side of (1.30) should be equal, i.e.
or, by a simple
ρ0 (z) = ρt (Tt z),
(1.31)
ρt (z) = ρ0 (T−t z).
(1.32)
hange of variables,
This is the way the density in phase spa e evolves in time. It simply says that the probability density of
mi rostate
may of
z
at time
t
is the same as the one that had the initial
ondition of
z
at the initial time. We
ourse nd a dierential equation for the probability density by just taking the time derivative
on both sides of (1.32). This gives
d
d
ρt (Tt z) = ρ0 (z) = 0.
dt
dt
(1.33)
that, again, expresses the fa t that as we move with the ow in phase spa e, the probability density
does not
hange. Further appli ation of the
hain rule leads to the Liouville equation for the probability
density in phase spa e,
∂t ρ(z, t) = −iLρ(z, t),
∂
∂t to denote the partial derivative with respe t to time. It is
onstru tion, that the formal solution of the Liouville equation (1.34) is given by (1.32).
where we have used the notation
obvious, by
(1.34)
∂t =
1.4 Equilibrium at the mi ros opi
level
15
An alternative derivation of the Liouville equation starts from the realization that the ow in phase
spa e is in ompressible. In order to prove this, note that from Hamilton's equations (1.2) we may dene
the ow velo ity in phase spa e as
v(z) = J0
∂H
(z)
∂z
(1.35)
Therefore, by taking the divergen e of this velo ity eld we obtain
∂2H
∂
v(z) = J0
(z) = 0
∂z
∂z∂z
The zero
omes from the fa t that we are
(1.36)
ontra ting a fully antisymmetri
matrix
J0
with a fully
symmetri one, the Hessian of the Hamiltonian. Eq. (1.36 tells us that the ow velo ity has null divergen e
in phase spa e. This is another ree tion of the fa t that the ow in phase spa e is in ompressible, or that
volumes are
onserved by the Hamiltonian dynami s. As any other probability density, the probability
in phase spa e satises a
The
ontinuity equation that ree ts the fa t that probability is lo ally
onserved.
ontinuity equation is
∂t ρt (z) = −
By using the in ompressibility
∂
·v(z)ρt (z)
∂z
(1.37)
ondition (1.36), we obtain again the Liouville equation (1.34).
1.4
Equilibrium at the mi ros opi
Let us
onsider now the nal state predi ted by the Liouville equation. A basi
under whi h
level
mathemati al question is
onditions the Liouville equation (1.34), whi h is a rst order partial dierential equation,
∂t ρ(z, t) = 0. That this is not generally the ase an be seen by
ρ(z, 0) = δ(z −z0 ) that expreses that we know with ertainty
ase, we know that the solution is given by ρ(z, t) = δ(Tt z0 − z), this is,
leads to a stationary solution with
onsidering an initial distribution of the form
that the initial state is
z0 .
In this
the distribution fun tion remains peaked at the solution of Hamilton's equations. There is no broadening
of the distribution fun tion and the system does not rea h a stationary state. However, if the dynami s
generated by the Hamiltonian is highly unstable (i.e.
haoti ), we may expe t that any non-delta initial
distribution will evolve with a sort of broadening. To be more spe i , if the dynami s of the system is
of the mixing type, then the system rea hes an ee tive stationary mi ros opi
probability that it is a
?
fun tion of the mi rostate only through the dynami al invariants of the system [ ℄. Usually, the proof
that a given system is of the mixing type is di ult but we will assume that our system is of the mixing
type and has, therefore, a tenden y to rea h a well dened equilibrium state.
Any distribution fun tion
ρ(z)
whi h is a fun tion
g(I(z))
will be, therefore, a stationary solution of
the Liouville equation and, as stated, we will assume that any stationary solution is of this type. This
ρeq (z). Therefore,
stationary distribution is alled the
equilibrium ensemble
lim ρ(z, t) = ρeq (z) = g(I(z)).
(1.38)
t→∞
Let us investigate the meaning of the fun tion
g(I)
by
onsidering the probability distribution
P eq (I)
of
dynami al invariants at equilibrium. By denition,
P eq (I) =
Z
dzρeq (z)δ(I(z) − I) =
where we have introdu ed the measure
Ωeq (I)
Z
dzg(I(z))δ(I(z) − I) = g(I)Ωeq (I),
of the region of phase spa e
(1.39)
ompatible with a given set
The Mi ros opi
16
Dynami s
of dynami al invariants
Z
eq
Ω (I) =
g(I)
Equation (1.39) allows to identify
dzδ(I(z) − I).
(1.40)
and (1.38) be omes
ρeq (z) =
P eq (I(z))
.
Ωeq (I(z))
(1.41)
Therefore, the equilibrium ensemble is fully determined by the probability distribution of dynami al
invariants at equilibrium.
It is obvious that the distribution of dynami al invariants at any time is itself invariant. The probability density of nding a value
I
of the dynami al invariants
P (I, t) =
Z
I(z)
is given by
dzρt (z)δ(I(z) − I)
(1.42)
The time derivative of this probability is
∂t P (I, t) =
Z
dz(−iL)ρt(z)δ(I(z) − I) =
Z
dzρt (z)iLδ(I(z) − I) = 0
Therefore, the probability of dynami al invariants is itself an invariant. As a
(1.43)
onsequen e the equilibrium
distribution of dinami al invariants is just the same as the initial distribution of dynami al invariants
P eq (I) = P (I, 0). We will denote P0 (I) = P (I, t) and then the equilibrium ensemble is just
ρeq (z) = ρ0
where we have introdu ed
ρ0
(1.44)
−N
with dimensions of (a tion)
in order to have
Ω0 (I) = ρ0
with the same physi al dimensions as
initial distribution
P0 (I(z))
.
Ω0 (I(z))
P0 (I) of dynami
P0 (I).
Z
dzδ(I(z) − I)
The equilibrium ensemble
(1.45)
ρeq (z) is fully determined on
e the
al invariants is known at the initial time. Eq. (1.44) is a fundamental
result of equilibrium Statisti al Me hani s. The intuitive meaning of (1.44) is very sugestive.
the measure of the submanifold of mi rostates
I
orresponding to the invariants
Ω0 (I)
is
and we may think
that it is proportional to the number of mi rostates that have a value I for the dynami invariants.
ρeq (z) of a given mi rostate z is the probability P0 (I(z)) of being in
Therefore, the probability density
the submanifold
I(z) = I
divided by the number of mi rostates of that submanifold. We
that at equilibrium all mi rostates with the same value of
I(z) are equiprobable.
of a mnemote hin al rule for (1.44) than a rigorous statement be ause, being a
the number of mi rostates satisfying
I(z) = I
Of
ould say, then,
ourse, this is more
ontinuum submanifold,
is innite.
The referen e value ρ0N
The fun tion
Ω(E)
and
energy shell.
ontains all the ma ros opi
The fun tion
thermodynami
in parti ular, are derived from this fun tion.
agree with
H(z) = E in phase spa
Ω(E) is usually termed the stru
gives the overall measure of the submanifold
ifold is usually termed the
?
e. This subman-
?
ture fun tion [ ℄
information about the system [ ℄. Equations of state,
It is also well-known that in order to have results that
orresponding results obtained dire tly from Quantum Me hani s, the appropriate value for
1.5 The single, all en ompassing problem of Non-Equilibrium Statisti al Me hani s
ρ0N
17
should be taken as
ρ0N =
Plan k's
onstant
h
1
h3N N !
(1.46)
gives the appropriate dimensions whereas the fa tor
?
bility of the parti les [ ℄ and it is known as the
orre t Boltzmann
ideal gas latter, it ensures the extensivity of the ma ros opi
of parti les that
the fa torial
N!
is due to the indistinguisha-
ounting. As it will be seen for the
entropy. Of
ourse, if we have a mixture
an be distinguished by some property (for example, they have dierent mass), then
oe ient
hanges a
ordingly to a
ount for the dierent equivalent ways of ordering the
parti les.
1.5
The single,
all en ompassing problem of Non-Equilibrium
Statisti al Me hani s
Non-Equilibrium Statisti al Me hani s is based on the fundamental presuposition that all ma ros opi
pro esses
an be ultimately understood in terms of the
far, we have seen how an isolated
lassi al laws of motion of isolated systems. So
lassi al system made of parti les is governed at the mi ros opi
level
by Hamilton's equations with a time-independent Hamiltonian and how any initial distribution over the
phase spa e will evolve towards the equilibrium ensemble.
This means that we restri t ourselves to
study the evolution towards the equilibrium state of an initial distribution whi h is not the stationary
solution of the Liouville equation
orresponding to the Hamiltonian of the system.
Our limitation on
isolated systems that de ay to equilibrium pre ludes, aparently, the possibility to study non-equilibrium
stationary states that are mantained with external
ouplings. These experimental situations, however,
do t into the framework of an isolated de aying system whenever we
in
onta t with reservoirs, in su h a way that the
onsider the system under study
omposed system of system+reservoirs is isolated. In
this view, a stationary state is just an extremely long-lived de ay towards the global equilibrium of the
system+reservoir, where the time s ale towards equilibrium is di tated by the size of the reservoir.
Therefore, the basi
pro ess that we study is how an arbitrary initial ensemble de ays towards the
equilibrium ensemble
ρ0 (z) −→ ρeq (z)
A system left to evolve will rea h the equilibrium state
Be ause no matter how an isolated system is prepared
(1.47)
orresponding to the Hamiltonian of the system.
1 it will go towards the same equilibrium state, the
equilibration of a system is a parti ularly simple way to prepare and
ontrol the initial state of a system
in the preparation phase of an experiment. Therefore, we assume that the initial ensemble
equilibrium state of
ertain Hamiltonian
transforms the original Hamiltonian
equilibrium ensemble of
the equilibrium state of
H1 and
H1 .
H0
H0 .
At
t=0
some parameter of the Hamiltonian
into another Hamiltonian
it will evolve a
H1 .
The ensemble
ording to the dynami s generated by
Callen in his magni ent book Thermodynami s states in a
ρ0 (z)
is the
hanges and
ρ0 is no longer
H1 until it rea
the
hes
rystalline senten e whi h is, perhaps,
the essential tenet of the book: The single, all en ompassing problem of thermodynami s is the determi-
nation of the equilibrium state that eventually results after the removal of internal
omposite system . If we think about this problem in mi ros opi
straint
an always be des ribed at a mole ular level as a
onstraints in a
losed,
terms, the removal of an internal
on-
hange of the Hamiltonian of the system and,
therefore, is a pro ess of the form des ribed in the previous paragraph. For this reason, we may state
that the fundamental problem of Thermodynami s is, indeed the very same problem of Non-Equilibrium
1 Provided
the distribution of dynami
invariants is the same in all preparations!
18
The Mi ros opi
Statisti al Me hani s, ex ept that in the latter
Dynami s
ase not only the nal equilibrium state is seeked for, but
the ri her question of how this state is rea hed in time is answered.
2
The Mesos opi Dynami s
2.1
Levels of des ription
The theory of
oarse-graining is a formalization of the pro ess of representing a given system with less
information than that
aptured by the a tual mi rostate of the system.
levels of des ription depending
may be des ribed at dierent
one retains ma ros opi ally.
One and the same system
on the amount of information whi h
The dierent levels of des ription of a system are
hara terized by the
I(z) and a set of phase fun tions A(z) whi h
A(z) that hara terize a given level of des ription
dynami al invariants of the system
are not dynami al
invariants. The phase fun tions
will be referred to
as
relevant variables
ma ros opi
but they have re eived in the past a number of dierent names. Ma rostates,
variables, gross variables,
olle tive variables,
oarse-grained variables, rea tion
order parameters, internal variables, stru tural variables, et .
oordinates,
are all synonyms for relevant variables.
A(z) we denote a olle tion of phase fun tions ea h one labeled with a dis rete index
A(z) = {Aµ (z), µ = 1, · · · , M }.
ation of the relevant variables A(z) is an art of the theory of oarse-graining and a ru ial
With the symbol
like in, for example,
The identi
element in order to des ribe ma ros opi ally a system with many degrees of freedom. As we have stressed
in the previous
hapter, we are
on erned with the transition (1.47) from an initial ensemble towards the
equilibrium ensemble. Usually, this happens in a way that it is possible to identify
patterns that emerge in the
of
olle tive motions and
ourse of the relaxation towards equilibrium. When we stop stirring our
oe, whi h at its most mi ros opi
level is made of
olliding atoms, vorti es are
up
learly visible that
suggest that the relaxation happens following paths in the phase spa e. These paths are
hara terized
by phase fun tions whose values evolve in time mu h slowly than other phase fun tions. We will see that
from a pra ti al point of view only when there is a
relevant variables and the
lear separation of time s ales between the sele ted
orrelations of their time rate of
hange, it is possible to have simple dynami
equations for the relevant variables. When this happens we have that the relevant variables forget their
past rapidly and their future is essentially determined by their present values. In these
ases, we say that
the des ription is Markovian. The general strategy when there is no su h a separation of time s ales is
to look for additional variables that also evolve in
omparable time s ales as the ones that we believe are
the slow variables. By enlarging the set of relevant variables, we hope that the resulting des ription may
be Markovian.
There are few guiding prin iples for the sele tion of relevant variables. Whenenever we have
onserved
or quasi onserved variables, we expe t that they will need to be in luded in the des ription. Therefore,
we will always in lude in the set of relevant variables the dynami
the Hamiltonian
H(z)
invariants of the system. In parti ular,
will be in luded in the des ription. If for some reason, we expe t a
hara teristi
feature (orientation, stre hing, elongation, et . ) to play a role in the dynami s of the system, then we
The Mesos opi
20
need to in lude the phase fun tions that best
Dynami s
apture su h feature in the
olle tion of relevant variables.
Ideally, one would like to develop tools for analyzing the ow in phase spa e and automati ally produ e
the appropiate relevant variables for the problem at hand. This pattern re ognition pro ess is far from
being addressed in the literature be ause the problems to fa e are enormous given the high dimensionality
of phase spa e.
Let us turn ba k to the motion of the system in phase spa e. We have said that the single en ompasing problem of Non-Equilibrium Statisti al Me hani s is the study of how the system approa hes the
equilibrium state. This pro es is one in whi h a
energy shell (or a
loud of points initially
on entrated in a region of the
loud inhomogenous in any other way) spread uniformly in that energy shell. Now,
imagine that the system has an additional invariant (you may think, for example, about total linear
momentum). This means that the ow in phase spa e will be stratied in layers, in whi h the points on
every layer never leaves the layer (in order to
do not have su h additional dynami
onserve the dynami al invariant). Now, imagine that we
invariant, but the ow in phase spa e is quasi-stratied, in a way
that it be omes rapidly homogeneous in layers and the ow from one layer to another o
this is the dynami al s enario at the mi ros opi
denes the layers will be slow variables and good
In the present
andidates to be relevant variables.
hapter, we want to derive the governing equations for the probability distributions
of the relevant variables starting from the mi ros opi
dynami
urs slowly. If
level, we expe t that the phase fun tion that impli itly
equation for the probability distribution
dynami s of the system.
P (a, t)
The resulting exa t
takes, in the Markovian approximation, the
form of a Fokker-Plan k equation (FPE). This FPE was introdu ed by Green in a seminal paper in 1952
by using a a line reasoning where he assumed that the sto hasti
Markovian sto hasti
pro ess of relevant variables was a
pro ess. From this assumption he derived the governing equation for the transition
probability of this pro ess.
Green's paper is arguably a
ornerstone in the theory of non-equilibrium
statisti al me hani s. The derivation by Zwanzig in 1961 of the same FPE with the help of a proje tion
operator, showed how this equation emerges in the limit of
lear separation of time s ales from an exa t
non-Markovian equation.
2.2
Sto hasti
We will
pro esses in Phase Spa e
onsider relevant variables
A(z) with the property that an equation like A(z) = a denes a proper
Ω(a) of the submanifold A(z) = a dened
Z
Ω(a) = dzρ0N δ(A(z) − a)
(2.1)
submanifold of phase spa e. By this we mean that the measure
as
exists and it is well dened. Roughly,
with a given ma rostate
a.
Ω(a)
ounts the number of mi rostates
des ribe the evolution of the relevant variables in terms of sto hasti
?
fully
hara terized [ ℄ by giving the hierar hy of
of having the value
a1
z
that are
ompatible
When the relevant variables dene proper submanifolds it is possible to
at time
t1
and the value
pro esses. A sto hasti
pro ess is
joint probability distributions P (a1 , t1 , · · · , an , tn )
a2
Let us express this joint probability in mi ros opi
at time
t2
et . where
t1 < t2 < · · · < tn ,
for all
n.
terms. In this se tion we assume that the number of
parti les is known. Extension to the ma ro anoni al phase spa e is straightforward. The motion in phase
spa e
i-th
Γ
of a mi rostate
parti le,
z = (q1 , . . . , qN , p1 , . . . , pN ),
an be viewed as a sto hasti
pro ess will be denoted by
where
qi , pi
pro ess itself. The
ρn (z1 t1 , . . . , zn tn ).
The
are the position and momentum of the
orresponding joint probabilities for this
ombination of the two fa ts, that the randomness is
given only at the initial time and that the later evolution is deterministi , makes the one-time probability
ρ1 (z, t) the most relevant joint probability of the hierar hy. In fa t, all the joint probabilities ρn
n ≥ 2 an be expressed in terms of the one-time probability density ρ1 (z, t). By denoting with Tt z
density
with
2.2 Sto hasti
pro esses in Phase Spa e
the solution of Hamilton's equations with initial
z
onditions
21
we have
ρn (z1 t1 , . . . , zn tn ) = ρ1 (z1 t1 )δ(z2 − Tt2 −t1 z1 ) . . . δ(zn − Ttn −tn−1 zn−1 )
(2.2)
The one-time probability density or ensemble density satises the well known Liouville's equation with
formal solution given by Eq.
un ertainty in the initial
We next
(1.32) All the sto hasti ity in the pro ess given by
onsider the evolution of the relevant variables
mi rostate itself. The values that these dynami
The
n-time
Tt z
arises from the
onditions.
A(Tt z) as a
variables take
onse uen e of the evolution of the
an be regarded as a sto hasti
pro ess.
P (a1 t1 , . . . , an tn ) whi h hara terize the sto hasti pro ess of the
probability densities ρ(z1 t1 , . . . , zn tn ) of the orresponding mi ros opi
joint probability densities
relevant variables and the n-time
pro ess are related to ea h other through
P (a1 t1 , . . . , an tn ) =
whi h
Z
...
Z
ρ(z1 t1 , . . . , zn tn )δ(A(z1 ) − a1 ) . . . δ(A(zn ) − an )dz1 . . . dzn
an be further simplied by using (2.2) and integrating over
P (a1 t1 , . . . , an tn ) =
Z
where it has been used that
has inverse
z1 = Tt1 z
Tt Tt′ = Tt+t′ .
P (a1 t1 , . . . , an tn ) =
Z
a
By performing the
ρ(z, 0)
hange of
oordinates
(2.4)
z = T−t1 z1
(whi h
an be written as
dzρ(z, 0)δ(A(Tt1 z) − a1 )δ(A(Tt2 z) − a2 ) . . . δ(A(Ttn z) − an )
This is the nal form for the ma ros opi
ma rostates
z2 , . . . , zn
dzρ(z, t1 )δ(A(z) − a1 )δ(A(Tt2 −t1 z) − a2 ) . . . δ(A(Ttn −t1 z) − an )
and unit ja obian) equation (2.4)
over the initial ensemble
(2.3)
(2.5)
n-time joint probability density whi h appears as an integral
of delta fun tions that ontra t the des ription from mi rostates
and whi h involves the mi ros opi
dynami s
z
to
Tt .
Initial ensemble
A basi
question that arises now is, what is the a tual fun tional form of
prin iple we
at a given
annot measure the initial mi ros opi
A(z).
parti ular distribution
P (a, 0),
with numeri al out omes
a
ess to the measurement of the
oarse-
over the system prepared in an identi al manner at the initial time.
P (a, 0).
However, there are many dierent
?
ρ(z, 0)
ρ(z, 0) with
the sole information that it should
Both distribution fun tions are related through
P (a, 0) =
Z
that
dzδ(A(z) − a)ρ(z, 0).
an produ e the same
(2.6)
P (a, 0).
ording to information theory [ ℄, the least biased distribution whi h is
P (a, 0)
As mentioned, in
whi h is the out ome of a repeated set of measurements of the fun tions
provide pre isely the distribution
information
ρ(z, 0)?
exa tly. If we are going to des ribe a system
In general, all the information we have about our system at the initial time is a
Therefore, we have to determine the distribution fun tion
A
z
oarse-grained level, we must assume that we have a
grained variables
A(z)
state
Whi h is the
orre t one?
ompatible with the ma ros opi
is the one that maximizes the entropy fun tional
S[ρ0 ] ≡ −kB
Z
Γ
ρ(z, 0) ln
ρ(z, 0)
dz.
ρ0N
(2.7)
The Mesos opi
22
onditioned to the restri tion (2.6).
Dynami s
We en ounter here a problem of Lagrange multipliers.
By intro-
λ(a)Rfor the ontinuum set
R of restri tions (2.6) (one for ea h a), we maximize the
tional I[ρ0 ] = S[ρ0 ] +
dxλ(a)P (a, 0) + µ dxP (a, 0), where the µ Lagrange multiplier stands for
normalization to unity restri tion of ρ(z, 0). The Lagrange multipliers are obtained by substituting
du ing the multipliers
fun
the
?
the maximum value into the restri tion (2.6). The following nal result is obtained [ ℄
ρ(z, 0) =
where
Ω(a)
P (A(z), 0)
,
Ω(A(z))
(2.8)
is given in (2.1).
By substituting this initial ensemble into equation (2.5) and by
hoosing
t1
equal to 0 for simpli ity,
one obtains
P1 (a1 , 0)
P (a1 0, . . . , an tn ) =
Ω(a1 )
We now introdu e the
t2
at time
and in
a3
Z
dzδ(A(z) − a1 )δ(A(Tt2 z) − a2 ) . . . δ(A(Ttn z) − an )
(2.9)
P (a1 0|a2 t2 , . . . , an tn ) of nding the system in a2
provided it was in a1 at the initial time t1 = 0. It is
onditional probability density
at time
t3
and so on till
a n , tn ,
given by
P (a1 0, a2 t2 , . . . , an tn )
=
P (a1 , 0)
Z
1
dzδ(A(z) − a1 )δ(A(Tt2 z) − a2 ) . . . δ(A(Ttn z − an )
=
Ω(a1 )
P (a1 0|a2 t2 , . . . , an tn ) ≡
For further referen e it is
probability
onvenient to
onsider
1
P (a1 0|a2 t2 ) =
Ω(a1 )
Z
n=2
(2.10)
in (2.10), that re eives the name of
transition
dzδ(A(z) − a1 )δ(A(Tt2 z) − a2 )
(2.11)
This is a fundamental equation that relates the transition probability of the CG variables with the miros opi
dynami s. It
number of mi rostates
ompatible with
of mi rostates
a1
an be given an heuristi
ompatible with
a1 that after a time are at
ompatible with
a1
interpretation as follows. The numerator
Ω(a1 )
is the
while the denominator in (2.11) is the number of mi rostates
a2 .
Therefore, the transition probability is just the fra tion
that after a given time are at
a2 .
2.3 Green's view of
2.3
Green's view of
oarse-graining
23
oarse-graining
Green in his 1952 remarkable paper presented the essentials of the theory of
The basi
assumption taken by Green and on whi h the whole
that the sto hasti
pro ess of the relevant variables is a
oarse-graining as we know it.
onstru tion of
oarse-graining is based is
ontinuum Markov pro ess. As we will see in this
se tion, this single hypothesis is su ient to obtain a dynami al equations for the relevant variables, the
Fokker-Plan k equation, with all the obje ts appearing in the dynam s dened in terms of mi ros opi
expressions. But rst we have to introdu e the Markov sto hasti
A Markov pro ess is hara terized by the fa t that the
pro ess and some of its properties.
n-time joint probability
an be fully expressed in
terms of the one-time probabability, and the transition probability. In fa t, any other
n-time
probability
is written as
P (a1 t1 , . . . , an tn ) = P (a1 , t1 )P (a1 , t1 |a2 , t2 ) · · · P (an−1 , tn−1 |an , tn )
For a Markov pro ess the full sto hasti
pro ess is
hara terized by the one time probability and the
transition probability alone. We have already en ountered a Markov pro ess, the one
the deterministi
Hamiltonian dynami s des ribed in Eq. (2.2). If we
P2 (a1 , t1 |a2 , t2 , a3 , t3 ) of
having
a2
at t2 and
a3
(2.12)
onsider the
at t3 provided that we had
a1
orresponding to
onditional probability
at t1 , the Markov property
states
P (a1 , t1 |a2 , t2 , a3 , t3 ) = P (a1 , t1 |a2 , t2 )P (a2 , t2 |a3 , t3 )
We
an interpret the Markov
(2.13)
ondition in geometri al terms as we did when dis ussing the mixing
a1 that happen to be
a1 that will be at a2 at t2 (irrespe tive of
a2 that will be in a3 at time t3 .
property. The Markov property says that the fra tion of points of the submanifold
at
a2
at t2 and then at
a3
at
t3
equals the fra tion of points of
where they will go afterwards) times the fra tion of points of
If we integrate Eq. (2.13) over
a2
we need to have the following
P2 (a1 , t1 |a3 , t3 ) =
This
onsisten y
ondition is known as the
Z
onsisten y
ondition
da2 P (a1 , t1 |a2 , t2 )P (a2 , t2 |a3 , t3 )
(2.14)
Chapman-Kolmogorov equation for the transition proba-
bilities. The intuitive idea with this equation is that the probability of a transition from
as the sum of all the transition probabilites over an intermediate state
a1
to
a3
is given
a2 .
The Chapman-Kolmogorov is an integral equation that links all the transtion probabilities of a Markov
pro ess.
as the
There exists an equivalent dierential form for the Chapman-Kolmogorov whi h is named
Fokker-Plan k equation.
The derivation of the Fokker-Plan k equation from the Champan-
?
Kolmogorov equation is presented in [ ℄ and we only quote the nal result. The Fokker-Plan k equation
governs the one time probability distribution
∂
1 ∂2
∂
P (a, t) = − D(1) (a)P (a, t) +
D(2) (a)P (a, t)
∂t
∂a
2 ∂a∂a
(2.15)
The two time joint probability and one time probability are related by
P (a1 , t1 ) =
Z
da0 P (a0 , t0 , a1 , t1 )
be ause both sides of this equation are the probability of nding
a0
at time
t0 .
a1
at time
(2.16)
t1
irrespe tive of the value of
This equation gives the following integral equation relating the one time probability and
The Mesos opi
24
Dynami s
the transition probability
P (a1 , t1 ) =
Z
da0 P (a0 , t0 )P (a0 , t0 |a1 , t1 )
Note that, from Eq. (2.17) if we take as initial
ondition
(2.17)
P (a0 , t0 ) = δ(a0 − â) then P (a, t) = P (â, t0 |a, t),
this is, the transition probability is identi al to the one-time probability with a Dira
tion. As a
delta initial
ondi-
onsequen e, the transition probability also satises the Fokker-Plan k equation (2.15)
∂
∂
1 ∂2
P (a0 t0 |a, t) = − D(1) (a)P (a0 t0 |a, t) +
D(2) (a)P (a0 t0 |a, t)
∂t
∂a
2 ∂a∂a
where, by denition, we have that the initial
(2.18)
ondition for this equation is
P (a0 , t0 |a, t0 ) = δ(a − a0 )
Therefore, Green's basi
assumption that the sto hasti
(2.19)
pro ess of the relevant variables is a Markov
pro ess is equivalent to the hypothesis that the one-time and transition probabilities of the relevant
variables obey the Fokker-Plan k equation.
Mole ular expresion of drift and diusion
The drift ve tor
D(1) (a)
and the diusion tensor
D(2) (a)
of moments of the transition probability at short times.
introdu ed in Eq.
(2.15) are given in terms
Its parti ular form is, a tually, spe ied as
?
onditions in the derivation of the Fokker-Plan k equation from the Chapman-Kolmogorov equation [ ℄.
However, we may also obtain the spe i
form of these obje ts by just the requirement that the transition
probability obeys the FPE (2.18). Let us see how this arise. The solution of the Fokker-Plan k equation
(1)
(a), D(2) (a) may depend in
(2.18) is di ult to obtain in general due to the fa t that the obje ts D
general on the state
a
in a non-linear way. Nevertheless, it is possible to obtain an expli it solution for
short times. Be ause the initial
that for su iently short times
peaked. In this
ondition (2.19) of the transition probability is a Dira
t = t0 + ∆t
ase, we may approximate in
results in a Fokker-Plan k equation with
delta, we expe t
∆t ≈ 0, the transition probability will remain highly
(1)
(2.18) D
(a) ≈ D(1) (a0 ) and D(2) (a) ≈ D(2) (a0 ). This
with
onstant
oe ients whi h is easy to solve
∂
∂
1 ∂2
P (a0 t0 |a, t) = −D(1) (a0 ) P (a0 t0 |a, t) + D(2) (a0 )
P (a0 t0 |a, t)
∂t
∂a
2 ∂a∂a
The exa t solution of this equation with initial
(2.20)
ondition (2.19) has a Gaussian form
1 −1
P (a0 , t0 |a1 , t0 + ∆t) = exp −
a1 − a0 − ∆tD(1) (a0 ) D(2)
(a0 ) a1 − a0 − ∆tD(1) (a0 )
2∆t
1
×
(2.21)
M/2
(2π∆t)
det(D(2) (a0 ))1/2
provided that the inverse of the diusion matrix
D(2) (a)
exists. In systems with inertia the inverse does
not exist but it is nevertheless still possible to write down the transition probability, that will in lude
?
some delta fun tions [ ℄. The transition probability (2.21) has the following moments
Z
Z
da1 (a1 − a0 )P (a0 , t0 |a1 , t0 + ∆t) = D(1) (a0 )∆t
da1 (a1 − a0 )(a1 − a0 )P (a1 , t0 + ∆t|a0 , t0 ) = D(2) (a0 )∆t + D(1) (a0 )D(1) (a0 )∆t2
(2.22)
2.3 Green's view of
The se ond moment
Z
oarse-graining
25
an also be expressed as
da1 (a1 − a0 − D(1) (a0 )∆t)(a1 − a0 − D(1) (a0 )∆t)P (a1 , t0 + ∆t|a0 , t0 ) = D(2) (a0 )∆t
D(1) (a)
Note that these expressions give the drift
and diusion
D(2) (a)
(2.23)
in terms of the rst and se ond
moments of the short time form of the transition probability. But note now that we have also a mi ros opi
expression for the transition probability, given in (2.11)! Therefore, by substituting (2.11) into (2.22) and
D(1) (a) and D(2) (a). Consider the rst term
(2.23) we may obtain expli it mole ular expressions for
D(1) (a0 ) and use Eq. (2.11) in Eq. (2.22)
D(1) (a0 ) =
1
∆t
Z
da(a − a0 )
a
We perform the integral over the variable
D
(1)
1
(a0 ) =
∆t
Z
1
Ω(a1 )
Z
δ(A(z) − a0 )δ(A(T∆t z) − a)dz
of the Dira
delta fun tion and obtain
δ(A(z) − a0 )
(A(T∆t z) − A(z)) =
dz
Ω(a0 )
where we have introdu ed the generalized mi ro anoni al average or
a0
h· · · i
Eq. (2.25) is the desired mi ros opi
≡
dz
A(∆t) − A(0)
∆t
a0
(2.25)
onditional average
δ(A(z) − a0 )
···
Ω(a0 )
expression for the drift
of (2.11) into (2.23) we obtain a mi ros opi
D(2) (a0 ) =
Z
(2.24)
D(1) (a0 ).
(2.26)
In a similar way, by substitution
expression for the diusion tensor
E a0
1 D
[A(∆t) − a0 − D(1) (a0 )∆t][A(∆t) − a0 − D(1) (a0 )∆t]
∆t
Einstein-Helfand formula
This expression known as the
for the diusion
(2.27)
oe ient and it basi-
ally says that the square displa ement of the relevant variables in reases proportional to
∆t,
with the
proportionality fa tor given by the diusion tensor.
Now, while both (2.25) and (2.27) are mi ros opi
depend on the time
∆t.
Of
to expand the above expressions in terms of
of
us
∆t.
expressions for the drift and diusion tensor, they
ourse we would like to eliminate somehow the dependen e on
However, the fa t that
∆t,
∆t
∆t.
We want
and keep only the zero order term whi h is independent
although small, is not vanishing small leads to some subtelties. Let
onsider the drift term rst. As a rst approximation, we would use a simple Taylor expansion of the
relevant variable
A(T∆t z) = A(z) + iLA(z)∆t + O(∆t)2
(2.28)
and negle t terms of high order, to obtain
a0
D(1) (a0 ) = hiLAi
The drift would be given as the
this is not quite
(2.29)
onditional average of the velo ity of the relevant variables. However,
orre t. Even though we have assumed that
go to zero stri tly be ause then the basi
hold. We need to
+ O(∆t)
∆t
is small, the time interval
∆t
annot
Markovian assumption with no memory of the past would not
onsider the time interval
∆t
as ne esarily nite and the mi ros opi
dynami s may
The Mesos opi
26
Dynami s
have time to do funny things. It is, therefore, preferable to use the following identity
A(T∆t z) − A(z) =
Z
∆t
d
dt A(Tt z) =
dt
0
Z
∆t
dtiLA(Tt z)
(2.30)
0
By using this identity in (2.25), we obtain an mathemati ally identi al expression for
D(1) (a0 ) =
1
∆t
Z
∆t
dt
0
Z
dz
1
δ(A(z) − a0 )
iLA(Tt z) =
Ω(a0 )
∆t
∆t
Z
D(1) (a0 ),
this is
dt hiLA(t)ia0
(2.31)
0
∆t of the average velo
Tt z → z , with unit Ja obian,
In this way, we see that it the drift is the time average over the time interval
ity
of the relevant variables.
and
We now perform the
hange of variables
obtain
D
Now
(1)
1
(a0 ) =
∆t
Z
∆t
dt
0
Z
dz
δ(A(T−t z) − a0 )
iLA(z)
Ω(a0 )
(2.32)
onsider the identity, similar to (2.30)
A(T−t z) = A(z) +
0
Z
dt′ iLA(Tt′ )
(2.33)
−t
D
We
(1)
1
(a0 ) =
∆t
Z
∆t
0
1
dt
Ω(a0 )
an now formally expand the Dira
Z
δ A(z) +
0
dt′ iLA(Tt′ z) − a0
−t
Z
Z
dzδ A(z) +
0
′
dt iLA(Tt′ z) − a0 iLA(z)
−t
delta fun tion around
= δ (A(z) − a0 ) −
(2.34)
A(z) − a0
∂
δ (A(z) − a0 )
∂a0
Z
0
dt′ iLA(Tt′ (z)) + · · ·
(2.35)
−t
By inserting this formal expansion into (2.34) we have
a
D(1) (a0 ) = hiLAi 0
Z ∆t
Z 0
Z
1
∂
1
+
dt
dt′ dzδ (A(z) − a0 ) iLA(Tt′ (z))iLA(z)
∆t 0
Ω(a0 ) ∂a0 −t
Z ∆t
Z 0
Z 0
Z
∂2
1
1
′
′′
dt
dt
dt
dzδ (A(z) − a0 ) iLA(Tt′′ (z))iLA(Tt′ (z))iLA(z)
+
∆t 0
Ω(a0 ) ∂a0 ∂a0 −t
−t
+ ···
This expression
(2.36)
an be written in a more
ompa t form as
a0
D(1) (a0 ) = hiLAi
+
∂
1
1
Ω(a0 )
Ω(a0 ) ∂a0
∆t
Z
∆t
0
1
∂2
1
+
Ω(a0 )
Ω(a0 ) ∂a0 ∂a0
∆t
+ ···
dt
Z
0
a0
dt′ hiLA(t′ )iLAi
−t
Z
∆t
dt
0
Z
0
−t
dt
′
Z
0
a0
dt′′ hiLA(t′′ )iLA(t′ )iLAi
−t
(2.37)
2.3 Green's view of
oarse-graining
27
By retaining all the terms in this expression we have an expression that is mathemati al equivalent to
a
(2.25). We observe, therefore, that, in addition to the naive term hiLAi 0 we have additional ontributions
that are time integrals of orrelation fun tions. The task now is to evaluate whether these
0
s ale as (∆t) or with a higher power of ∆t. Let us onsider the rst orrelation matrix
1
∆t
Z
∆t
dt
0
Z
0
′
a0
′
dt hiLA(t )iLAi
−t
1
=
∆t
=
1
∆t
1
=
∆t
where we have performed a
order to have a proper
1
∆t
Z
∆t
dt
0
Z
Z
∆t
dt
0
Z
∆t
Z
dt′′
0
Z
t
ontributions
a0
dt′′ hiLA(−t′′ )iLAi
0
Z
∆t
a0
dt hiLA(−t′′ )iLAi
t′′
∆t
a0
dt′′ (∆t − t′′ ) hiLA(−t′′ )iLAi
(2.38)
0
hange of variables. Next we add and substra t the term
hiLAia0 hiLAia0
in
orrelation that de ays to zero
0
′
a0
′
dt hiLA(t )iLAi
=
Z
∆t
dt′′ h(iLA(−t′′ ) − hiLAia0 )(iLA − hiLAia0 )i
a0
0
−t
Z ∆t
1
a
dt′′ t′′ h(iLA(−t′′ ) − hiLAia0 )(iLA − hiLAia0 )i 0
−
∆t 0
∆t
(2.39)
+
hiLAia0 hiLAia0
2
a0
a
′′
a
We expe t that the matrix of orrelations h(iLA(−t ) − hiLAi 0 )(iLA − hiLAi 0 )i
de ays to zero as
′′
|t | in reases and that the integral then onverges to a term whi h is independent of ∆t, for su iently
large
∆t.
We denote with
D(a0 )
D(a0 ) ≡
Z
the resulting matrix
∆t
a0
dt′′ h(iLA(−t′′ ) − hiLAia0 )(iLA − hiLAia0 )i
(2.40)
0
On the other hand, we also expe t that the se ond integral in Eq.
the prefa tor
1/∆t,
this se ond integral will de ay as
1/∆t1 .
(2.39) also
Therefore, if
∆t
onverges, but due to
is slightly larger than the
orrelation time in whi h the velo ity u tuations of the relevant variables de ay, we may negle t the
se ond integral. Finally, the last term in Eq. (2.39) is of order
∆t.
The analysis of the third and higher order terms in Eq. (2.37) be omes readily very
will assume, though, that these terms are small and that
ompli ated. We
an be negle ted. This approximation is not
rigorous and one has to judge the resulting expression a posteriori. Therefore, we obtain the following
form for the drift term
a0
D(1) (a0 ) = hiLAi
+
1
∂
Ω(a0 )D(a0 ) + O(∆t)
Ω(a0 ) ∂a0
where the expli itly displayed terms in this equation are independent of
∆t.
Note that, as
(2.41)
ompared with
the naive result (2.29), we have an additional term in the drift. This term arises due to the fa t that the
time
∆t is large
ompared with the
At the same time,
∆t
orrelation of the u tuations of the velo ity of the relevant variables.
needs to be short in
omparison with the evolution of the relevant variables
themselves, otherwise, we are not allowed to use the Gaussian form for the transition probability on
whi h the whole pro edure is bases. The existen e of su h time
∆t
is based on the fa t that there must
be a separation of time s ales between the relevant variables and the u tuations of their velo ities.
1 One
may
onsider a simple example in whi h the
orrelation de ay as exponentials to get a feeling for this behaviour.
The Mesos opi
28
Dynami s
We may obtain an expression for the diusion tensor (2.27) by expanding again in the small time
∆t,
In order to have the proper orders in
(A(T∆t z) − a0 ) by
substitute just one of the terms
∆t.
using (2.28),
this is
D
E a0
D(2) (a0 ) = [iLA − D(1) (a0 )][A(∆t) − A(0)]
+ O(∆t)
This expression is equivalent to (2.27) up to small terms of order
diusion tensor by mi ros opi
means.
∆t
and
(2.42)
an also be used to obtain the
This se ond form of the diusion tensor has no spe i
name
asso iated to it.
Still, a third way to express the diusion tensor is by using the identity (2.30) in (2.42). Then we
have
D(2) (a0 ) =
Z
∆t
0
D
E a0
dt [iLA − D(1) (a0 )]iLA(t)
+ O(∆t)
We may symmetrize this expression by substra ting
D
(2)
(a0 ) =
Z
0
This is the
∆t
D(1) (a0 )
to the last
(2.43)
iLA(Tt z)
term
D
E a0
dt [iLA − D(1) (a0 )][iLA(t) − D(1) (a0 )]
+ O(∆t)
(2.44)
Green-Kubo formula for the diusion tensor. The three expressions (2.27),(2.42) and (2.44)
are equivalent forms for the diusion tensor in mi ros opi
We may now ask about the value of
terms.
We have used the form (2.21) for the transition probability
annot take the mathemati al limit ∆t → 0
D(2) (a) = 0. Clearly ∆t has to be a time
(1)
whi h is large enough for the orrelations of the u tuations iLA − D
of the velo ity to have de ayed
(1)
or, equivalently, for the square of the displa ement A(∆t) − a0 − D
(a0 )∆t in the Einstein-Helfand
under the assumption that
∆t
∆t.
be ause the above mi ros opi
is small.
However, we
expressions give the result
expression to have rea hed a behaviour linear in time. At the same time, it has to be short enough for
being able to use the Gaussian approximation for the transition probability. As this Gaussian behaviour
(1)
(a), D(2) (a) with D(1) (a0 ), D(2) (a0 ), we
is related to the fa t that we ould make the substitution D
expe t that the time
∆t
has to be short in front of the typi al s ale of evolution of the relevant variables.
In summary, one obvious requirement for the validity of the above expressions is that there exists a
lear
separation of time s ales between the relevant variables and the the u tuations of the velo ity of the
relevant variables.
In other words, the relevant variables need to be slow variables (in the time s ale
of its velo ity u tuations).
the dynami
It is obvious that one needs to in lude in the set of relevant variables
a
invariants, in parti ular the Hamiltonian, of the system as they are the slowest possible
variables. This is, the rst
A(z)
needs to be the Hamiltonian. In this way the
ontain a Dira
delta fun tion over the Hamiltonian whi h is
omponent of the ve tor
onditional expe tations (2.26) do, in fa t,
nothing else that the equilibrium mi ro anoni al ensemble.
In this se tion, we have presented the theory of
oarse-graining as was given by Green. In prin iple,
and quite often in pra ti e, this is all what we would need to know about the theory of
Of
ourse, the whole
onstru tion is based on the assumption that the sto hasti
oarse-graining.
pro ess of relevant
variables has the Markov property. We have seen that this assumption is equivalent to postulate that
the one-time and transition probabilities obey the Fokker-Plan k equation. If this is true, then we have
expli it mole ular expressions for the dynami s of the relevant variables.
Of
ourse, in the pro ess of
obtaining these mole ular expressions we have followed a number of non-rigorous approximations that
may leave us with an un onfortable feeling.
In order to get some more insight into the problem, in the next se tions, we will not postulate the
Fokker-Plan k equation but rather will derive it from the mi ros opi
that in this way, we will have some more light into the problem.
Hamilton's equations. We hope
2.4 Zwanzig view of
2.4
Zwanzig view of
2.4.1
The evolution of the mi ros opi
distribution
P (a, t)
ensemlbe
P (a, t),
ρ(z, t) a
operator te hnique.
fun tion
F (z)
Z
ρ(z, t).
This
?
ording to
δ(A(z) − a)ρ(z, t)dz.
losed dynami al equation for
ourse, we would like to have a
underlying dynami s given by
ording to the Liouville equation indu es an evolution
be ause both are related a
P (a, t) =
Of
29
oarse-graining
Exa t equation for
of the mesos opi
oarse-graining
losed equation
P (a, t)
(2.45)
that makes no referen e to the
an be obtained with the help of a proje tion
Following Zwanzig [ ℄, we introdu e a proje tion operator
of phase spa e
Γ
P
that applies to any
PF (z) = hF iA(z) ,
where we have introdu ed the
onditional average
hF ia =
1
Ω(a)
P on an
A(z). The
Z
hF ia
(2.46)
by
dzρ0N δ(A(z) − a)F (z),
(2.47)
Note that the ee t of the operator
arbitrary fun tion of phase spa e is to transform it into a
P satises the proje tion property P 2 = P . We
2
omplementary proje tion operator Q = 1 − P whi h satises PQ = 0 and Q = Q.
fun tion of the relevant variables
introdu e also the
The operators
P, Q
operator
satisfy
Z
for arbitrary fun tions
dzρ0N A(z)PB(z) =
A(z), B(z).
It is
Z
dzρ0N B(z)PA(z),
(2.48)
onvenient to introdu e the following notation
Ψa (z) = δ(A(z) − a),
and
(2.49)
onsider the Dira 's delta fun tion as an ordinary phase fun tion with a ontinuum index
to the formal solution (1.15) this phase fun tion will evolve a
a.
A
ording
ording to
Ψa (Tt z) = exp{iLt}Ψa(z),
(2.50)
∂t Ψa (Tt z) = exp{iLt}iLΨa(z).
(2.51)
and, therefore,
Now we introdu e a mathemati al identity between operators
exp{iLt} = exp{iLt}P +
Z
t
dt′ exp{iLt′ }PiLQ exp{iLQ(t − t′ )} + Q exp{iLQt}.
(2.52)
0
This identity
derivative and
an be proved by taking the time derivative on both sides. If two operators have the same
oin ide at
t=0
then they are the same operator. We now apply this identity (2.52) to
the left hand side of (2.51). After some algebra, whi h uses the expli it form of the operators
properties (2.48), and the
P, Q,
the
hain rule in the form
iLΨa (z) = iLAµ
∂
Ψa (z),
∂aµ
(2.53)
The Mesos opi
30
Dynami s
where summation over repeated indi es is implied, one obtains,
∂
·vµ (a)Ψa (Tt z)
∂aµ
Z t
Z
∂
∂ Ψa′ (Tt′ z)
+
dt′ da′ Ω(a′ )
·Dµν (a, a′ , t − t′ )· ′
∂a
∂a
Ω(a′ )
µ
j
0
+ Q exp{iLQt}QiLΨa(Tt z).
∂t Ψa (Tt z) = −
We have dened the drift
vµ (a)
and the diusion tensor
′
P (a, t)
′
h(iLAν − hiLAν ia ) exp{iLQt}Ψa(iLAµ − hiLAµ ia )ia .
Dµν (a, a′ , t) =
for
through
hiLAµ ia ,
vµ (a) =
If we multiply (2.54) by
Dµν (a, a′ , t)
ρ(z, 0), integrate over z , and use (??), we obtain a nal exa
∂
·vµ (a)P (a, t) +
∂t P (a, t) = −
∂aµ
Z
t
dt
0
′
Z
da′ Ω(a′ )
Z
t and
(2.55)
losed equation
∂ P (a′ , t′ )
∂
·Dµν (a, a′ , t − t′ )· ′
,
∂aµ
∂a ν Ω(a′ )
where we have used that the initial ensemble (2.8) is a fun tion of
A(z)
(2.56)
and, therefore,
dzρ(z, 0)Q exp{iLQt}QiLΨa(Tt z) = 0,
where we have used the hermiti ity (2.48) as well as the proje tion property
2.4.2
(2.54)
(2.57)
Qf (A) = 0.
The Markovian approximation and the Fokker-Plan k Equation
Equation (2.56) is an exa t an rigorous
losed equation governing the distribution fun tion
P (a, t).
No
approximations have been made and, essentially, it is another way of rewriting the Liouville equation.
In prin iple, it is as di ult to solve as the original Liouville equation. However, as it happens often
in Physi s, just by rewriting the same thing in a dierent form, it is possible to perform suitable approximations that allow for an advan e in the understanding of the problem. In the
approximation is
ase of (2.56), the
alled the Markovian approximation and transforms the integro-dierential equation
into a simple Fokker-Plan k equation.
The Markovian assumption is one about separation of time s ales between the time s ale of evolution
of the phase fun tion
A(z)
and the rest of variables of the system. If this separation of time s ales exists
Dµν (a, a′ , t−t′ ) de ays, the probability P (a, t′ ) has not hanged
then, in the time s ale in whi h the tensor
appre iably. S hemati ally, we write the memory term in (2.56) as
Z
t
dt′ D(t − t′ )P (t′ ) ≈ P (t)
Z
∆t
D(t′ )dt′ .
(2.58)
0
0
This approximation is depi ted in Fig. 2.1. We have extended in (2.58) the upper limit of integration
to a time ∆t su iently large for the memory kernel D(t) to have de ayed . Note that the tensor
Dµν (a, a′ , t − t′ ) is a quantity of order (iLA)2 , i.e. se ond order in the time derivatives of the relevant
variables. The time s ale of evolution of
P (a, t)
is the same as the time s ale of the variables
A(z).
The
approximation (2.58) amounts, therefore, to negle t third order time derivatives of the relevant variables
in front of se ond order terms.
We, therefore,
onsistently perform a formal expansion of the tensor
2.4 Zwanzig view of
oarse-graining
31
P(t)
P(t’)
D(t)
D(t−t’)
t
t
t’=0
t’
Figure 2.1: The Markovian approximation.
Dµν (a, a′ , t − t′ )
in (2.55) in terms of
iLA
and keep only se ond order terms. Then,
exp{iLQt}ΨaQiLA = Ψa exp{iLQt}QiLA + O(iLA)2 .
O(iLA3 )
Therefore, up to terms of order
(2.59)
we have
Dµν (a, a′ , t) = δ(a − a′ )h(iLAν − hiLAν ia ) exp{iLQt}(iLAµ − hiLAµ ia )ia ,
and the tensor be omes diagonal in
a, a′ .
(2.60)
By substitution of the approximate form (2.60) into the exa t
equation (2.56) and using (2.58) we obtain
∂t P (a, t) = −
∂
∂
∂ P (a, t)
vµ (a)P (a, t) +
Ω(a)Dµν (a)
,
∂aµ
∂aµ
∂aν Ω(a)
(2.61)
where we have dened the diusion tensor
Dµν (a) =
Z
∆t
dt′ h(iLAν − hiLAν ia ) exp{iLQt′ }(iLAµ − hiLAµ ia )ia ,
(2.62)
0
Note that within the negle t of third order terms, we
an also substitute the proje ted dynami s with
the real dynami s, this is
Dµν (a) =
Z
∆t
dτ h(iLAν − Vν (a))) exp {−iLτ } (iLAµ − Vµ (a))i
Ea
(2.63)
0
We summarize for
ompleteness the rest of quantities appearing in (2.61)
vµ (a)
h. . .ia
Ω(a)
As we in lude in the set
A(z)
= hiLAµ ia ,
Z
1
=
dzρ0N δ(A(z) − a) . . . ,
Ω(a)
Z
=
dzρ0N δ(A(z) − a).
the total energy as a relevant variable, it is
(2.64)
onvenient to single out its
The Mesos opi
32
Dynami s
ee t. Let us write expli itly
vµ (a, E) =
h. . .ia
=
Ω(a, E) =
It is
hiLAµ ia,E ,
Z
1
dzρ0N δ(H(z) − E)δ(A(z) − a) . . . ,
Ω(a, E)
Z
dzρ0N δ(H(z) − E)δ(A(z) − a).
onvenient to multiply the numerator and denominator of the
Ω(E) ≡
Z
(2.65)
onditional averages with
dzδ(H(z) − E)
(2.66)
whi h is the measure of the number of mi rostates of a given energy. In this way, we obtain the
onditional
averages as
h · ia,E =
1
P mic (a)
Z
dzρmic (z)δ(A(z) − a) · · ·
where we have introdu ed the usual mi ro anoni al ensemble
ρmic (z)
(2.67)
and the equilibrium probability of
the relevant variables as
1
δ(H(z) − E)
Ω(E)
Z
P eq (a) = dzρmic (z)δ(A(z) − a)
ρmic (z) =
(2.68)
at the same time we have
Ω(a, E) = Ω(E)P eq (a)
Be ause
iLH = 0, many of the
(2.69)
terms in the FPE vanish, pre isely those involving derivatives with respe
to the total energy. For this reason, we may write (2.61) as
∂t P (a, t) = −
∂
∂ eq
∂ P (a, t)
Vµ (a)P (a, t) +
P (a)Dµν (a)
,
∂aµ
∂aµ
∂aν P eq (a)
(2.70)
whi h may also be written in the equivalent form
∂t P (a, t) = −
∂S
∂
∂
∂
Vµ (a) + Dµν (a)
P (a, t)
Dµν (a)
(a) P (a, t) +
∂aµ
∂a
∂a µ
∂aν
where we have introdu ed the entropy fun tion
S(a)
throught the
(2.71)
elebrated Einstein's formula for u -
tuations
P eq (a) = exp{S(a)/kB }
eq
We an he k that P (a) is the equilibrium solution. By substituting
eq
P (a) will be a stationary solution of the FPE (2.61) if and only if
X ∂
vµ (a)P eq (a) = 0.
∂a
µ
µ
(2.72)
P eq (a)
into (2.61) we have that
(2.73)
2.4 Zwanzig view of
By using the denition (2.55), the
∂
vµ (a)P eq (a) =
∂aµ
Z
oarse-graining
onstrained average (2.47), and the
dzρmic (z)
∂
δ(A(z) − a)iLAµ (z) = −
∂aµ
Z
33
hain rule we have
dzρmic (z)iLδ(A(z) − a) = 0,
iLH = 0.
where we have integrated by parts the Liouville operator and used that
Equation (2.61) is the desired Fokker-Plan k equation for the dynami s at the mesos opi
grained level of des ription. This equation is one of the
?
(2.74)
oarse-
ornerstones of non-equilibrium statisti al me-
?
hani s and was obtained by Zwanzig in 1961 [ ℄ following the path pioneered by Green in [ ℄. In this
equation, all the obje ts
vµ (a), Dµν (a, t)
and
Ω(a)
have a denite mi ros opi
denition. In parti ular,
(2.62) is a general form of the well-known Green-Kubo formulae that relates the transport
Dµν (a)
with a time integral of a
whenever there is a
orrelation fun tion of mi ros opi
lear separation of time s ales su h that the Markovian approximation is valid.
Clearly, the FPE will des ribe
orre tly the evolution of
the typi al time s ales of de ay of the
equation shorter time s ales.
time.
oe ients
variables. The FPE (2.61) is valid
P (a, t) only
Dµν (a).
orrelation involved in
for times whi h are larger than
We
For this short time s ales, the transport
annot investigate with this
oe ients start to depend on
34
The Mesos opi
Dynami s
3
Example: Diusing intera ting olloidal
parti les
At the most mi ros opi level, we an model the
olloidal suspension by assuming that the solid suspended
obje ts are spheri al and we need only 6 degrees of freedom for des ribing the state of the obje t, the
position
Qi
and the momentum
Pi
of its
enter of mass. For irregular obje ts we would need also to
onsider orientation, angular velo ities, et . The uid in whi h these solid olloidal parti les are suspended
will be des ribed at the most mi ros opi
of the mole ules that
mi ros opi
level by the positions
onstitute the uid.
state will be denoted by
qi
and momenta
pi
of the
enter of mass
Again, we assume spheri al mole ules for simpli ity.
z = {qi , pi , Qi , Pi }.
The
The evolution of the mi rostate is governed by
Hamilton's equations,
∂H(z)
,
∂Pi
∂H(z)
Ṗi = −
,
∂Qi
∂H(z)
,
∂pi
∂H(z)
ṗi = −
,
∂qi
Q̇i =
q̇i =
(3.1)
where the Hamiltonian is given by
H(z) =
Here,
mi
X p2
1 X SS
P2
i
+
Vij (q) + VijSC (q, Q) + VijCC (Q) .
+ i
2mi 2Mi
2 ij
i
is the mass of a solvent mole ule,
Mi
the mass of a
olloidal parti le, and
are the potential of the for es between solvent mole ules, solvent and
(3.2)
V SS , V SC , V CC
olloidal parti les, and
olloidal
parti les, respe tively.
In prin iple, the dierential equations (3.1)
an be solved numeri ally with a
is known as mole ular dynami s and allows us to keep tra k of all the mi ros opi
?
[ ℄. The smallest typi al time s ale is a
omputer. The te hnique
dynami s of the system
ollision time in the range of pi ose onds and,
onsistently, we will
need to use a time step for the numeri al solution whi h is mu h smaller than this time s ale. However,
if the mass of the
olloidal parti les is mu h larger than the mass of the solvent parti les, as it o
reality, the evolution of the
olloidal parti les will be very slow in
solvent mole ules. If we are interested in the motion of the
enormous number of time steps (and, therefore, of
the
olloidal parti les, then we would need an
omputer time) to observe an appre iable motion of
olloidal parti les. To study these large time s ales in a
absolutely impra ti able.
urs in
omparison with the evolution of the
olloidal suspension, mole ular dynami s is
Example: Diusing intera ting
36
olloidal parti les
We illustrate now how the general formalism developed in the previous se tion
ase of a
an be applied to the
olloidal suspension in order to derive the FPE. The idea is simply to translate to our system the
dierent obje ts dened in (2.62), (2.65) that appear in the FPE (2.61). The mi ros opi
given in (3.2). We sele t as relevant variables
take numeri al values
Qi .
Let us
A(z) = x
the positions of the
Z
dzρmic (z)
Y
mi ro anoni al ensemble in order to
δ(Qi − Qi )
ompute averages. The
equilibrium distribution of the positions of the
dq is a
(3.4)
eff
E.
By using the
T.
The parti ular value of
T
is
anoni al ensemble, we may write the
olloidal parti les as
P mic (Q) ∝ exp{−βV CC (Q)}
R
anoni al ensemble instead of the
anoni al ensemble is given
is proportional to the inverse of the temperature
xed in order to give an average energy given by
≡ exp{−βV
(3.3)
1
exp{−βH(z)},
Z
ρeq (z) =
where
whi h
i
As we know from equilibrium statisti al me hani s, we may use the
β = (kB T )−1
Qi
onsider the equilibrium probability for these variables. It is given by
P mic (Q) =
where
Hamiltonian is
olloidal parti les
Z
dq exp{−β V CS (Q, q) + V SS (q) }
(Q)},
(3.5)
ondensed notation for the integral over solvent positions. We have introdu ed the ee tive
potential as
V
eff
(Q) = V CC (Q) − kB T ln
The ee tive potential has a
additional
ontribution
Z
V CC (Q)
After performing the integrals over the Dira
delta fun tions, the
1
Ω(Q)
h· · · iQ =
Note that this
and equilibrium averaged solvent mediated
olloidal parti les.
takes the form
in whi h the
(3.6)
oming from the dire t intera tion potential and an
ontribution that represents the ee t of the stati
intera tion between
Z
onstrained average in (2.65) now
dqρeq (q, Q) · · · .
(3.7)
onstrained average is simply an equilibrium average over the solvent degrees of freedom,
olloidal parti les are assumed to be xed at the values
ensemble average in whi h the
Q.
olloidal parti les a t as external stati
Be ause the time derivatives
is now the
dq exp{−β V CS (Q, q) + V SS (q) }.
iLA
are simply
Pi /Mi ,
the drift term
onstrained equilibrium average of the momentum of the
It is, therefore, an equilibrium
for e elds.
v(a) = hiLAia
dened in (2.65)
olloidal parti les, whi h is zero by
isotropy of the equilibrium ensemble. The diusion tensor (2.62) be omes
Dij (Q) =
Z
∆t
dt′ hVj exp{iLt′ }Vi iQ .
(3.8)
0
The nal FPE (2.61) takes now the form
"
#
eff
∂
∂V (Q)
∂
∂
∂t P (Q, t) =
βDij (Q)
P (Q, t) +
Dij (Q)·
P (Q, t),
∂Qi
∂Qj
∂Qi
∂Qj
In this example, we observe how the general Fokker-Plan k des ription
an be applied to a spe i
(3.9)
level
37
of des ription of a given system.
get the stru ture of the
The essential benets of this approa h are that it is very simple to
oarse-grained equation. Also, we obtain expli it mi ros opi
expressions for all
the obje ts in the FPE. In parti ular, the diusion tensor whi h des ribes the mutual, solvent-mediated
inuen e of the
of the
olloidal parti les is given in terms of the auto and
ross- orrelations of the velo ities
olloidal parti les, where the averages are taken over the solvent degrees of freedom whi h are
distributed a
ording to an equilibrium ensemble in the presen e of the external elds due to the stati
olloidal parti les. The FPE that governs now the probability density
equation.
P (Q, t) is
alled the Smolu howsky